Matrix Cheat Sheet

Given, $A$ is a square matrix.

Determinant (in-general)
$\text{Det}(A) = \sum_{k=1}^{n} a_{ik} (-1)^{k+i} \text{Det}(A(i,k))$
$A_{i_{1}j_{1}}A_{i_{2}j_{2}}\cdots A_{i_{n}j_{n}}\epsilon_{j_{1}\cdots j_{n}} = \text{Det}(A)\epsilon_{i_{1}\cdots i_{n}}$
Matrix Derivative
$\frac{d A^{-1}}{dx} = - A^{-1} \frac{dA}{dx} A^{-1}$
Matrix exponential
$e^A = \sum_{k=0}^{\infty} \frac{1}{k!} A^k$
$e^A = \lim_{m \to \infty} \left(1 + \frac{1}{m} A \right)^m$
$\text{Det}(e^{A})= e^{\text{Tr}(A)}$
Matrix logarithm
$A^{x} = e^{x \ln(A)}$ where $x \in \mathbb{R}$

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Published on May 7, 2021

Last revised on Jul 8, 2021